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Singapore MOE · Primary 6 · Mathematics

Fractions

Revision notes · 3 parts · 2026-07-16
1

Dividing Proper Fractions

This section teaches you how to divide one proper fraction by another. You will learn the key rule of 'Keep, Change, Flip' and how to simplify your answers. Mastering these skills is important for solving problems involving parts of a whole.

❓ Before you read — have a guess: When you divide fractions, what do you do to the second fraction?

You flip it over to find its reciprocal.

⚡ Key points — the 6 things to remember
  • A proper fraction has a numerator smaller than its denominator.
  • Division of fractions is the same as multiplying by the reciprocal of the divisor.
  • The reciprocal of a fraction is found by flipping the numerator and denominator.
  • Always invert the divisor (the second fraction) before multiplying.
  • Simplify fractions by cross-cancellation only after changing division to multiplication.
  • The final answer should always be in its simplest form.

What you must be able to do

  • Divide a proper fraction by a proper fraction.

What is Dividing Proper Fractions?

Dividing proper fractions means finding out how many times one proper fraction fits into another proper fraction, or sharing a fractional amount into smaller fractional parts. A proper fraction is a fraction where the top number (numerator) is smaller than the bottom number (denominator), like 1/2 or 3/4.

  • The number being divided is called the dividend.
  • The number that divides the dividend is called the divisor.
  • The answer to a division problem is called the quotient.
0 1/4 1/2 3/4 1 Dividend (1/2) Divisor (1/4)
Visualising 1/2 ÷ 1/4. You can see two 1/4 segments in 1/2.

When you divide fractions, you are essentially asking 'How many groups of the divisor can fit into the dividend?' For example, if you divide 1/2 by 1/4, you are asking 'How many 1/4s are there in 1/2?'

💡 Exam Tip Always identify the dividend (first fraction) and the divisor (second fraction) correctly before starting any calculation.
📍 Singapore Focus Imagine you have 1/2 of a chocolate bar. If you want to cut it into pieces that are each 1/4 of the original bar, how many pieces would you get? This is a 'measurement' type of division problem.

The Reciprocal Rule for Division

To divide fractions, you 'Keep, Change, Flip'. This means you keep the first fraction (dividend) as it is, change the division sign to a multiplication sign, and flip the second fraction (divisor) to find its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator.

  • The reciprocal of a/b is b/a.
  • Multiplying by the reciprocal is the same as dividing.
a/b ÷ c/d = a/b × d/c
a/bThe dividend (first proper fraction)
c/dThe divisor (second proper fraction)
d/cThe reciprocal of the divisor
Worked example

A baker has 3/4 kg of flour. If each small cake needs 1/8 kg of flour, how many small cakes can he bake?

  1. Write the division problem: 3/4 ÷ 1/8
    Why this step? This sets up the problem based on the question.
  2. Keep the first fraction, change division to multiplication, flip the second fraction: 3/4 × 8/1
    Why this step? This is the 'Keep, Change, Flip' rule for dividing fractions.
  3. Multiply the numerators and denominators: (3 × 8) / (4 × 1) = 24/4
    Why this step? This performs the multiplication after applying the rule.
  4. Simplify the improper fraction to a whole number: 24 ÷ 4 = 6
    Why this step? This converts the improper fraction to its simplest form, giving the final answer.
6 cakes
✏️ Now you try

You have 2/5 of a bottle of juice. If each serving is 1/10 of a bottle, how many servings can you get?

  1. Write the division problem. 2/5 ÷ 1/10
  2. Apply the 'Keep, Change, Flip' rule. 2/5 × 10/1
  3. Multiply the numerators and denominators. (2 × 10) / (5 × 1) = 20/5
  4. Simplify the answer. 4
Reveal the answer 4 servings
⚠️ Common Mistake Students often invert the first fraction (dividend) instead of the second fraction (divisor) when dividing. — Correct: Always invert only the divisor (the fraction after the division sign). The dividend stays the same.
💡 Exam Tip Remember 'KCF': Keep (first fraction), Change (division to multiplication), Flip (second fraction).
💡 Exam Tip Double-check that you have inverted the correct fraction before multiplying.
🧠 Memory Aid KCF: Keep, Change, Flip
📍 Singapore Focus This rule is used when you need to find out how many smaller fractional parts are contained within a larger fractional amount, like cutting a length of ribbon into smaller pieces.

Simplifying Fractions During Division

After you have changed a division problem into a multiplication problem using the 'Keep, Change, Flip' rule, you can simplify the fractions before multiplying. This is called cross-cancellation, and it makes the numbers smaller and easier to work with.

  • You can only cross-cancel after changing the division to multiplication.
  • Cross-cancellation involves dividing a numerator and a denominator by a common factor.
Worked example

You have 2/3 of a cup of sugar. If a recipe calls for 4/9 of a cup of sugar, how many times can you make the recipe?

  1. Write the division problem: 2/3 ÷ 4/9
    Why this step? This sets up the problem based on the question.
  2. Apply 'Keep, Change, Flip': 2/3 × 9/4
    Why this step? This applies the rule for dividing fractions, turning it into a multiplication problem.
  3. Cross-cancel: Divide 2 (numerator) and 4 (denominator) by 2. Divide 3 (denominator) and 9 (numerator) by 3.
    Why this step? This step simplifies the numbers before multiplying, making the calculation easier.
  4. Multiply the new numerators and denominators: (1 × 3) / (1 × 2) = 3/2
    Why this step? This performs the multiplication with the simplified numbers.
  5. Convert the improper fraction to a mixed number or simplify if possible: 1 1/2
    Why this step? This converts the improper fraction to a mixed number, which is a more common way to express the answer in context.
1 1/2 times

Cross-cancellation helps you avoid multiplying large numbers and then simplifying a large fraction. It's a shortcut to get to the simplest form faster. Remember, you can only cross-cancel diagonally across the multiplication sign, or vertically within a single fraction.

⚠️ Common Mistake Students sometimes try to 'cancel' or simplify terms across the division sign before inverting the divisor. — Correct: You must always invert the divisor and change the operation to multiplication first. Only then can you cross-cancel.
💡 Exam Tip Always perform 'Keep, Change, Flip' first. Cross-cancellation comes after you have a multiplication problem.
💡 Exam Tip Look for common factors between any numerator and any denominator after changing to multiplication.
📍 Singapore Focus This skill is useful in problems where you need to calculate how many times a certain fractional quantity can be made or measured from another fractional quantity, and simplifying early saves time.

🪤 Watch for these traps

The wrong answers examiners plant to catch you — and how to dodge them.

Tempting answer1/2 ÷ 3/4 = 2/1 × 3/4 = 6/4 = 1 1/2
Looks right because The student remembered to flip a fraction and multiply, but flipped the first fraction (dividend) instead of the second (divisor).
But actually The rule is to 'Keep' the first fraction, 'Change' the sign, and 'Flip' only the second fraction (the divisor). So, 1/2 ÷ 3/4 should be 1/2 × 4/3.
✓ How to avoid it Always write down 'Keep, Change, Flip' or 'KCF' to remind yourself which fraction to flip. The dividend (first fraction) always stays the same.
Tempting answer2/3 ÷ 4/9 = (2÷2)/(3÷3) ÷ (4÷2)/(9÷3) = 1/1 ÷ 2/3 = 1 × 3/2 = 1 1/2
Looks right because The student tried to simplify the fractions before applying the division rule, or cancelled across the division sign.
But actually Simplification (cross-cancellation) can only be done after you have changed the division to multiplication and inverted the divisor. You cannot cancel across a division sign.
✓ How to avoid it Follow the steps strictly: 1. Keep, Change, Flip. 2. Cross-cancel (if possible). 3. Multiply. 4. Simplify.

🔍 Question Decoder

What exam phrasings are really asking for.

When the question says… It is testing… Your move
How many [fractional part] are there in [fractional whole]? Quotitive division (measurement) Divide the 'fractional whole' by the 'fractional part'.
[Fractional amount] is shared equally among [fractional number] parts. How much does each part get? Partitive division (sharing) Divide the 'fractional amount' by the 'fractional number of parts'.
Find the quotient of [fraction A] and [fraction B]. Direct division of fractions Calculate [fraction A] ÷ [fraction B].

🧭 Understand → Plan → Work → Check

How to answer: Word problems involving division of fractions

  1. 1Understand the problem: What is given? What needs to be found? 1 mark
  2. 2Plan the solution: Choose the correct operation (division) and identify the dividend and divisor. Consider drawing a model. 1 mark
  3. 3Work out the solution: Show all steps clearly, including 'Keep, Change, Flip', cross-cancellation, and multiplication. 1-2 marks
  4. 4Check the answer: Is the answer reasonable? Does it make sense in the context of the problem? Is it in the simplest form? 1 mark
Example: A piece of cloth is 4/5 metre long. How many smaller pieces, each 1/10 metre long, can be cut from it?
✅ Earns the mark

“Understand: Given: Total cloth length = 4/5 m. Length of each small piece = 1/10 m. Find: Number of small pieces. Plan: This is a grouping problem, so I need to divide the total length by the length of one small piece. Operation: Division. Work: 4/5 ÷ 1/10 = 4/5 × 10/1 (Keep, Change, Flip) = (4 × 10) / (5 × 1) (Multiply numerators and denominators) = 40/5 = 8 Check: If each piece is 1/10 m, then 8 pieces would be 8 × 1/10 = 8/10 = 4/5 m. This matches the total length. The answer is reasonable and in simplest form. Answer: 8 smaller pieces can be cut.”

This answer clearly shows all four steps of the framework. It identifies the given information and what needs to be found, explains the chosen operation, shows clear working with units, and verifies the answer. It earns full marks.

❌ Loses the mark

“4/5 ÷ 1/10 = 4/5 × 10/1 = 40/5 = 8. Answer: 8 pieces.”

This answer only shows the 'Work' step and the final answer. It does not demonstrate understanding of the problem, planning, or checking. While the calculation is correct, it would lose marks for not showing the full problem-solving process, especially in higher-order thinking questions.

🗝️ Words that earn marks

Keep, Change, Flip Explaining the method for dividing fractions.
Reciprocal of the divisor Referring to the inverted second fraction in division.
Cross-cancellation Describing the simplification process before multiplying fractions.
Simplest form Ensuring the final fraction cannot be reduced further.
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2

Dividing with Whole Numbers and Mixed Numbers

In Primary 6, you will learn to divide whole numbers and mixed numbers by proper fractions, and vice versa. This skill builds on your knowledge of multiplying fractions and understanding reciprocals. Remember that dividing by a fraction is the same as multiplying by its reciprocal. This topic is crucial for solving real-world problems involving sharing, grouping, and finding unknown quantities.

❓ Before you read — have a guess: What happens to a whole number when you divide it by a proper fraction (a fraction less than 1)?

The result will be larger than the original whole number.

⚡ Key points — the 5 things to remember
  • To divide by a fraction, multiply by its reciprocal.
  • Always convert a whole number into a fraction (e.g., 5 = 5/1) before dividing.
  • Always convert a mixed number into an improper fraction before dividing.
  • Simplify fractions to their simplest form after multiplication.
  • Estimate your answer to check if it is reasonable, especially when dividing by fractions or mixed numbers.

What you must be able to do

  • Divide a whole number by a proper fraction.
  • Divide a proper fraction by a whole number.
  • Divide a mixed number by a proper fraction.
  • Divide a proper fraction by a mixed number.

Dividing a Whole Number by a Proper Fraction

This is when you find out how many parts of a certain fractional size are contained within a whole number quantity. For example, how many 1/2-metre pieces can be cut from a 5-metre rope.

  • A whole number can be written as a fraction by placing it over 1 (e.g., 4 = 4/1).
  • The reciprocal of a fraction is found by flipping the numerator and the denominator (e.g., the reciprocal of 1/4 is 4/1 or 4).
  • To divide by a fraction, you multiply by its reciprocal.
Worked example 🧰 Draw a model

A ribbon is 3 metres long. How many pieces of ribbon, each 1/4 metre long, can be cut from it?

  1. Convert the whole number to a fraction: 3 = 3/1.
    Why this step? We convert the whole number to a fraction to make it easier to perform fraction multiplication.
  2. Find the reciprocal of the divisor (1/4): The reciprocal is 4/1.
    Why this step? We find the reciprocal because dividing by a fraction is the same as multiplying by its reciprocal.
  3. Multiply the dividend (3/1) by the reciprocal of the divisor (4/1): 3/1 × 4/1.
    Why this step? We multiply the first fraction (dividend) by the reciprocal of the second fraction (divisor) to complete the division operation.
  4. Calculate the product: (3 × 4) / (1 × 1) = 12/1.
    Why this step? We multiply the numerators together and the denominators together to get the new fraction.
  5. Simplify the answer: 12.
    Why this step? We simplify the improper fraction to a whole number for the final answer.
12 pieces
✏️ Now you try

How many 1/3-litre servings can be poured from a 2-litre bottle of juice?

  1. Convert the whole number to a fraction. 2 = 2/1
  2. Find the reciprocal of the divisor (1/3). The reciprocal is 3/1.
  3. Multiply the dividend by the reciprocal of the divisor. 2/1 × 3/1
  4. Calculate the product and simplify. 6/1 = 6
Reveal the answer 6 servings
Ribbon Length Divided into Pieces Total Ribbon 1/4 m 1/4 m 1/4 m 3 m 3 m
Each metre of ribbon can be cut into 4 pieces of 1/4 metre. So, 3 metres can be cut into 3 × 4 = 12 pieces.
⚠️ Common Mistake Students often invert the first fraction (the dividend) instead of the second fraction (the divisor) when dividing. — Correct: Always invert only the divisor (the fraction you are dividing by) to find its reciprocal, then multiply.
💡 Exam Tip To avoid inverting the wrong fraction, remember 'Keep, Change, Flip': Keep the first fraction, Change the division sign to multiplication, Flip the second fraction (divisor) to its reciprocal.
🧠 Memory Aid KCF: Keep, Change, Flip for dividing fractions.
📍 Singapore Focus This type of problem is called 'quotitive division' where you are finding out how many groups of a certain size (1/4 m) are in a total quantity (3 m).

Dividing a Proper Fraction by a Whole Number

This involves sharing a fractional quantity equally among a whole number of groups. For example, sharing 3/4 of a cake among 3 friends.

  • The procedure is the same: convert the whole number to a fraction, find the reciprocal of the divisor, then multiply.
  • When a proper fraction is divided by a whole number (greater than 1), the result will always be a smaller proper fraction.
Worked example 🧰 Draw a model

3/4 of a pizza is shared equally among 3 children. What fraction of the whole pizza does each child get?

  1. Convert the whole number to a fraction: 3 = 3/1.
    Why this step? We convert the whole number to a fraction to prepare it for fraction multiplication.
  2. Find the reciprocal of the divisor (3/1): The reciprocal is 1/3.
    Why this step? We find the reciprocal because division by a fraction is equivalent to multiplication by its reciprocal.
  3. Multiply the dividend (3/4) by the reciprocal of the divisor (1/3): 3/4 × 1/3.
    Why this step? We multiply the first fraction by the reciprocal of the second fraction to perform the division.
  4. Multiply the numerators and denominators: (3 × 1) / (4 × 3) = 3/12.
    Why this step? We multiply across the numerators and denominators to get the product.
  5. Simplify the fraction: 3/12 = 1/4.
    Why this step? We simplify the fraction to its simplest form to give the final answer.
1/4 of the pizza
✏️ Now you try

A baker uses 2/5 of a bag of flour to make 4 identical loaves of bread. What fraction of the bag of flour is used for each loaf?

  1. Convert the whole number to a fraction. 4 = 4/1
  2. Find the reciprocal of the divisor (4/1). The reciprocal is 1/4.
  3. Multiply the dividend (2/5) by the reciprocal of the divisor (1/4). 2/5 × 1/4
  4. Calculate the product and simplify. 2/20 = 1/10
Reveal the answer 1/10 of the bag
Pizza Shared Equally Whole Pizza 3/4 Each Child 1/4 ?
The pizza is divided into 4 equal parts, and 3 parts are available. Sharing these 3 parts among 3 children means each child gets 1 part, which is 1/4 of the whole pizza.
💡 Exam Tip Always simplify your fraction to its simplest form at the end of the calculation. This is often a mark awarded in exams.
📍 Singapore Focus This is an example of 'partitive division', where you are sharing a quantity into a known number of equal parts.

Dividing a Mixed Number by a Proper Fraction

This involves finding how many fractional parts are in a quantity that includes both whole units and a fraction. A mixed number is a number made up of a whole number and a proper fraction (e.g., 2 1/2). An improper fraction is a fraction where the numerator is greater than or equal to the denominator (e.g., 5/2).

  • Before dividing, you must convert the mixed number into an improper fraction.
  • The steps for division remain the same: multiply by the reciprocal of the divisor.
Worked example

A baker has 2 1/2 kg of flour. If each cake needs 1/4 kg of flour, how many cakes can he bake?

  1. Convert the mixed number (2 1/2) to an improper fraction: (2 × 2 + 1) / 2 = 5/2.
    Why this step? We convert the mixed number to an improper fraction because direct division with mixed numbers is not allowed; all fractions must be in improper form for multiplication/division.
  2. Find the reciprocal of the divisor (1/4): The reciprocal is 4/1.
    Why this step? We find the reciprocal of the divisor to change the division problem into a multiplication problem.
  3. Multiply the improper fraction (5/2) by the reciprocal of the divisor (4/1): 5/2 × 4/1.
    Why this step? We multiply the dividend by the reciprocal of the divisor to complete the division.
  4. Multiply the numerators and denominators: (5 × 4) / (2 × 1) = 20/2.
    Why this step? We multiply the numerators and denominators to get the product.
  5. Simplify the fraction: 20/2 = 10.
    Why this step? We simplify the improper fraction to a whole number for the final answer.
10 cakes
✏️ Now you try

A piece of wood is 3 3/4 metres long. How many pieces, each 1/2 metre long, can be cut from it?

  1. Convert the mixed number (3 3/4) to an improper fraction. 15/4
  2. Find the reciprocal of the divisor (1/2). 2/1
  3. Multiply the improper fraction by the reciprocal of the divisor. 15/4 × 2/1
  4. Calculate the product and simplify. 30/4 = 15/2 = 7 1/2
Reveal the answer 7 1/2 pieces
0 1/2 1 1 1/2 2 2 1/2 3 2 1/2 kg flour
Visualising 2 1/2 kg of flour and dividing it into 1/4 kg portions. Each whole kg has four 1/4 kg portions. 2 kg has 8 portions. The remaining 1/2 kg has two 1/4 kg portions. Total = 8 + 2 = 10 portions.
💡 Exam Tip Students often forget to convert the mixed number to an improper fraction first. Always make this your first step when you see a mixed number in a division problem.
📍 Singapore Focus This is another example of quotitive division, where you are finding out how many groups of a certain size (1/4 kg) are in a total quantity (2 1/2 kg).

Dividing a Proper Fraction by a Mixed Number

This involves dividing a proper fraction by a quantity that is greater than 1 (a mixed number). The result will always be a proper fraction, smaller than the original dividend.

  • Just like before, the mixed number must be converted to an improper fraction first.
  • Then, find the reciprocal of the improper fraction (which was the mixed number) and multiply.
Worked example

A jug contains 3/4 litre of juice. If each serving is 1 1/2 litres, what fraction of a serving is in the jug?

  1. Convert the mixed number (1 1/2) to an improper fraction: (1 × 2 + 1) / 2 = 3/2.
    Why this step? We convert the mixed number to an improper fraction to allow for standard fraction multiplication/division procedures.
  2. Find the reciprocal of the divisor (3/2): The reciprocal is 2/3.
    Why this step? We find the reciprocal of the divisor to transform the division into a multiplication.
  3. Multiply the dividend (3/4) by the reciprocal of the divisor (2/3): 3/4 × 2/3.
    Why this step? We multiply the dividend by the reciprocal of the divisor to solve the problem.
  4. Multiply the numerators and denominators: (3 × 2) / (4 × 3) = 6/12.
    Why this step? We multiply the numerators together and the denominators together.
  5. Simplify the fraction: 6/12 = 1/2.
    Why this step? We simplify the resulting fraction to its simplest form.
1/2 of a serving
✏️ Now you try

A small bottle holds 2/3 cup of syrup. If a large recipe requires 2 1/2 cups of syrup, what fraction of the large recipe can be made with the small bottle?

  1. Convert the mixed number (2 1/2) to an improper fraction. 5/2
  2. Find the reciprocal of the divisor (5/2). 2/5
  3. Multiply the dividend (2/3) by the reciprocal of the divisor (2/5). 2/3 × 2/5
  4. Calculate the product and simplify. 4/15
Reveal the answer 4/15 of the large recipe
💡 Exam Tip Always check if your answer is reasonable. When you divide a proper fraction by a mixed number (which is greater than 1), your answer should be a smaller fraction than the original proper fraction. If it's larger, you likely made a mistake (e.g., inverted the wrong fraction).
📍 Singapore Focus This problem asks what fraction of a larger quantity (1 1/2 litres) is represented by a smaller quantity (3/4 litre).

🪤 Watch for these traps

The wrong answers examiners plant to catch you — and how to dodge them.

Tempting answerAn answer that is larger than the original proper fraction when dividing by a mixed number.
Looks right because Students might mistakenly think that division always makes numbers smaller, or they might invert the wrong fraction, leading to a larger product.
But actually When you divide a proper fraction (less than 1) by a mixed number (greater than 1), the result must be a smaller fraction. Dividing by a number greater than 1 always reduces the original quantity.
✓ How to avoid it Estimate your answer before calculating. If you are dividing a small number by a large number, expect a small answer. If your calculated answer is larger, recheck your steps, especially the reciprocal.

🔍 Question Decoder

What exam phrasings are really asking for.

When the question says… It is testing… Your move
How many [fractional part] are in [whole number/mixed number]? Division of whole/mixed number by a proper fraction (quotitive division). Convert whole/mixed number to improper fraction, then multiply by the reciprocal of the fractional part.
[Fractional quantity] is shared equally among [whole number] people/groups. How much does each get? Division of a proper fraction by a whole number (partitive division). Convert the whole number to a fraction (e.g., 3 = 3/1), then multiply the fractional quantity by its reciprocal (e.g., 1/3).
[Fractional quantity] is [fraction] of the total. What is the total? Working backwards to find the total quantity given a fractional part. Divide the fractional quantity by the given fraction (e.g., if 3/4 is 1/2 of total, then 3/4 ÷ 1/2).

🧭 Understand → Plan → Work → Check

How to answer: Multi-step word problems involving fractions.

  1. 1Understand: Read the problem carefully. What is given? What do you need to find? 1 mark
  2. 2Plan: Decide which operations to use. Can you draw a model to help? (e.g., bar model) 1 mark
  3. 3Work: Show all your working clearly, step-by-step. Include units. 2 marks
  4. 4Check: Is your answer reasonable? Does it make sense in the context of the problem? Write down your final answer statement. 1 mark
Example: A baker has 2 1/2 kg of flour. If each cake needs 1/4 kg of flour, how many cakes can he bake?
✅ Earns the mark

“Understand: Given: Total flour = 2 1/2 kg, Flour needed per cake = 1/4 kg. Find: Number of cakes. Plan: This is a grouping problem (quotitive division). I need to divide the total flour by the flour needed per cake. I will convert the mixed number to an improper fraction first. Work: 1. Convert 2 1/2 to an improper fraction: 2 1/2 = (2 × 2 + 1) / 2 = 5/2 kg 2. Divide total flour by flour per cake: 5/2 ÷ 1/4 3. Multiply by the reciprocal of the divisor: 5/2 × 4/1 4. Calculate the product: (5 × 4) / (2 × 1) = 20/2 = 10 Check: If each cake needs 1/4 kg, then 4 cakes need 1 kg. Since the baker has 2 1/2 kg, he can make 2 × 4 + (1/2 × 4) = 8 + 2 = 10 cakes. The answer is reasonable. Answer: The baker can bake 10 cakes.”

This answer clearly follows all steps of the framework. It shows the conversion, the division as multiplication by reciprocal, and simplification. Units are included in the working. The check step confirms the reasonableness of the answer, and a clear answer statement is provided. This earns full marks.

❌ Loses the mark

“2 1/2 ÷ 1/4 = 5/2 × 4 = 10 cakes.”

This answer shows the correct calculation but lacks the 'Understand' and 'Plan' stages. The 'Work' section is too brief, missing the step of converting 4 to 4/1, which is important for clarity. There is no 'Check' step or a proper answer statement. While the final answer is correct, the lack of clear working and framework steps might lose marks in an exam for presentation and understanding of the problem-solving process.

🗝️ Words that earn marks

Convert to improper fraction You see a mixed number in a calculation, especially before multiplication or division.
Multiply by the reciprocal You are performing division with fractions. This is the key step.
Simplify to simplest form You have completed a calculation and need to present the final fraction answer.
Whole number as fraction over 1 You need to include a whole number in a fraction calculation (e.g., 5 becomes 5/1).
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3

Solving Multi-Step Fraction Word Problems

This section teaches you how to solve word problems that involve several steps and different fraction operations. You will learn to handle scenarios where a fraction of a remaining quantity is used, and how to work backwards to find a total amount when only a part is known. Mastering these skills is key to tackling complex fraction questions in your exams.

❓ Before you read — have a guess: In a fraction problem, if you are told '1/2 of the remainder', what does 'remainder' refer to?

The amount or quantity that is left after a part has already been used or taken away.

⚡ Key points — the 6 things to remember
  • The word 'of' in fraction problems means to multiply.
  • When a problem mentions 'fraction of the remainder', you must calculate the fraction of what was left, not the original total.
  • Use a bar model to visualise fractions and relationships in word problems, especially for 'remainder' scenarios.
  • To find a total quantity when given a fractional part, you can work backwards by finding the value of one unit.
  • Break down multi-step problems into smaller, logical steps to solve them systematically.
  • Always convert mixed numbers to improper fractions before performing multiplication or division.

What you must be able to do

  • Solve word problems with many steps involving fractions, including those about what is left.

Understanding 'Of' and 'Remainder'

In fraction problems, the word of means to multiply. The remainder is the amount or quantity that is left after a part has been used or taken away.

  • To find a fraction 'of' a quantity, you multiply the fraction by the quantity.
  • If a fraction of a whole is used, the remainder is found by subtracting the used fraction from 1 (the whole).
  • When a problem says 'fraction of the remainder', you must calculate the fraction of what was left, not the original total amount.
Worked example 🧰 Draw a model

Ali spent 1/3 of his money on a book and 1/2 of the remainder on a pen. What fraction of his money was left?

  1. Fraction of money remaining after buying the book: 1 - 1/3 = 2/3
    Why this step? We subtract the fraction spent on the book from the whole (1) to find the portion of money Ali still had.
  2. Fraction of original money spent on pen (1/2 of the remainder): 1/2 × 2/3 = 1/3
    Why this step? We multiply the fraction for the pen (1/2) by the remaining fraction (2/3) to find what fraction of the original money was spent on the pen.
  3. Total fraction of money spent: 1/3 (book) + 1/3 (pen) = 2/3
    Why this step? We add the fraction spent on the book and the fraction spent on the pen to find the total fraction of money Ali used.
  4. Fraction of money left: 1 - 2/3 = 1/3
    Why this step? We subtract the total fraction spent from the whole (1) to find the final fraction of money Ali had left.
1/3 of his money was left.
✏️ Now you try

Mei Ling used 1/4 of her flour for cookies. She then used 1/3 of the remaining flour for a cake. What fraction of her original flour was left?

  1. What fraction of flour was left after making cookies? 1 - 1/4 = 3/4
  2. What fraction of the original flour was used for the cake? (Hint: 1/3 of the remainder) 1/3 × 3/4 = 1/4
  3. What total fraction of flour did Mei Ling use? 1/4 (cookies) + 1/4 (cake) = 2/4 = 1/2
  4. What fraction of her original flour was left? 1 - 1/2 = 1/2
Reveal the answer 1/2 of her original flour was left.
Ali's Money Problem Total Money Book (1/3) Pen (1/2 of remainder) Left Original Money
This bar model shows Ali's money divided into 3 equal units. 1 unit was spent on the book. Of the 2 remaining units, 1 unit was spent on the pen. 1 unit was left.
⚠️ Common Mistake In 'fraction of a remainder' problems, I should calculate the fraction of the original total instead of the remaining amount. — Correct: You must calculate the fraction of the amount that was left after the first part was used. For example, if 1/3 was spent, the remainder is 2/3. Then, '1/2 of the remainder' means 1/2 of 2/3.
💡 Exam Tip Always underline keywords like 'remainder' or 'left' in the question. This reminds you to use the correct base amount for the next step of calculation.
💡 Exam Tip Draw a bar model for 'fraction of a remainder' problems. It helps you visually see what portion is being referred to.
📍 Singapore Focus This concept is used in scenarios involving remaining quantities, such as spending money in stages, using ingredients from a supply, or consuming a portion of a resource over time.

Working Backwards to Find the Total

To work backwards means to use a known part of a quantity and its fraction to find the original total quantity.

  • If a fraction (e.g., 3/5) represents a certain amount (e.g., 18 pupils), you can find the value of one unit (1/5) by dividing the amount by the numerator (18 ÷ 3).
  • Once you know the value of one unit, multiply it by the denominator to find the total (value of 1 unit × 5).
  • Alternatively, you can divide the given amount by the fraction (e.g., 18 ÷ 3/5) to find the total.
Worked example 🧰 Draw a model

3/5 of the pupils in a class are girls. If there are 18 girls, how many pupils are there in total?

  1. Understand that 3 units represent 18 girls.
    Why this step? We identify that the numerator of the fraction (3) corresponds to the given number of girls (18).
  2. Find the value of 1 unit: 18 ÷ 3 = 6 pupils
    Why this step? We divide the number of girls by the number of units they represent to find the quantity for a single unit.
  3. Find the total number of units (5 units): 6 × 5 = 30 pupils
    Why this step? We multiply the value of one unit by the total number of units (the denominator, 5) to find the total number of pupils in the class.
There are 30 pupils in total.
✏️ Now you try

2/7 of the fruits in a basket are apples. If there are 14 apples, how many fruits are there in total?

  1. How many units represent the 14 apples? 2 units
  2. What is the value of 1 unit? 14 ÷ 2 = 7 fruits
  3. How many units represent the total number of fruits? 7 units
  4. What is the total number of fruits? 7 × 7 = 49 fruits
Reveal the answer There are 49 fruits in total.
Pupils in a Class Total Pupils Girls (18 pupils) Boys ? pupils Total Pupils
This bar model shows the class divided into 5 equal units. 3 units represent the 18 girls. We need to find the value of all 5 units.
💡 Exam Tip When working backwards, always check if your final answer makes sense. The total quantity should always be larger than the fractional part you started with.
💡 Exam Tip Remember that dividing by a fraction is the same as multiplying by its reciprocal. For example, 18 ÷ 3/5 is the same as 18 × 5/3.
📍 Singapore Focus This skill is essential for problems where you need to calculate the total quantity given a fractional part, such as finding the total number of students in a school or the original amount of money someone had.

Solving Multi-Step Word Problems

Multi-step word problems require you to perform more than one operation (addition, subtraction, multiplication, or division) with fractions to find the final answer.

  • Read the problem carefully to understand all the information given and what the question is asking you to find.
  • Break down the problem into smaller, manageable steps. Solve one step at a time.
  • Use models, such as bar models, to help visualise the fractions and the relationships between different quantities.
  • Always convert mixed numbers to improper fractions before you multiply or divide them. This makes calculations easier and prevents errors.

Solving multi-step fraction problems involves careful reading, planning, and executing calculations in the correct order. It often combines the skills of understanding 'of' and 'remainder' with working backwards to find totals.

💡 Exam Tip Students often rush and miss keywords like 'remainder' or 'left'. Always underline these words to remind yourself to use the correct base amount for the next step.
💡 Exam Tip Show all your working clearly, step-by-step. Even if you make a small calculation error, you can still get method marks if your steps are logical and easy to follow.
🧠 Memory Aid UPWC: Understand, Plan, Work, Check – a good strategy for all word problems.
📍 Singapore Focus These problems often involve real-life scenarios like budgeting money, sharing food, or measuring ingredients, where quantities are dealt with in multiple stages.

🪤 Watch for these traps

The wrong answers examiners plant to catch you — and how to dodge them.

Tempting answerWhen finding the total amount from a fractional part (e.g., 3/5 of total is 18), multiplying the amount by the fraction (18 × 3/5).
Looks right because Multiplication is often associated with 'of' or finding a part, and you might think you are finding a part of 18.
But actually If a part (18) is given and you need to find the whole (total), you must divide the given amount by the fraction (18 ÷ 3/5) or find the value of one unit and multiply by the total units.
✓ How to avoid it Always draw a bar model. If 3 units represent 18, then 1 unit is 18 ÷ 3. The total (5 units) is 1 unit × 5. This visual method helps prevent multiplying by the fraction incorrectly.
Tempting answerIncorrectly adding or subtracting fractions without a common denominator in multi-step problems.
Looks right because You might forget the rule for adding/subtracting fractions when focused on the word problem's complexity.
But actually Fractions can only be added or subtracted directly if they have the same denominator. If not, you must find a common denominator first.
✓ How to avoid it Before adding or subtracting, always check if the denominators are the same. If they are different, convert the fractions to equivalent fractions with a common denominator.

🔍 Question Decoder

What exam phrasings are really asking for.

When the question says… It is testing… Your move
What fraction of the remainder was used? Understanding 'fraction of a remainder' concept. First, find the fraction that represents the remainder. Then, multiply that remainder fraction by the given fraction.
How much money did [person] have at first? Ability to work backwards from a known part to find the original total. Determine what fraction of the total the known amount represents. Then, use that to find the value of one unit, and finally the total.
What fraction of the total was left? Ability to track changes to a whole quantity and find the final remaining fraction. Calculate the total fraction used or spent. Then, subtract this total from 1 (representing the whole) to find the fraction left.

🧭 Understand → Plan → Work → Check

How to answer: Any multi-step word problem involving fractions.

  1. 1Understand: Read the problem carefully. What information is given? What is the question asking for? Underline key numbers and words. 1 mark
  2. 2Plan: Choose a strategy. Will you draw a bar model? What operations will you use? Write down your plan. 1 mark
  3. 3Work: Show all your calculations step-by-step. Carry units through your working. Convert mixed numbers to improper fractions if needed. 2-3 marks
  4. 4Check: Does your answer make sense? Is it reasonable? Did you answer the question asked? Write a clear final answer statement with units. 1 mark
Example: Ali spent 1/3 of his money on a book and 1/2 of the remainder on a pen. He had $12 left. How much money did Ali have at first?
✅ Earns the mark

“Understand: Ali spent money in two steps, then had $12 left. I need to find his starting money. Plan: I will use a bar model to represent the money. I will work forwards to find the fraction of money left, then work backwards from the $12 to find the total. Work: 1. Fraction of money left after book: 1 - 1/3 = 2/3 2. Fraction of original money spent on pen: 1/2 × 2/3 = 1/3 3. Total fraction of money spent: 1/3 (book) + 1/3 (pen) = 2/3 4. Fraction of money left: 1 - 2/3 = 1/3 5. If 1/3 of Ali's money = $12 6. Total money (3/3) = $12 × 3 = $36 Check: If Ali had $36, he spent 1/3 × $36 = $12 on the book. Remainder = $36 - $12 = $24. He spent 1/2 × $24 = $12 on the pen. Money left = $24 - $12 = $12. This matches the problem. The answer is reasonable. Answer: Ali had $36 at first.”

This answer earns full marks because it clearly follows the 'Understand → Plan → Work → Check' framework. All steps are shown, calculations are correct, units are included, and the final answer is stated clearly. The 'Check' step confirms the solution.

❌ Loses the mark

“1/3 + 1/2 = 5/6 1 - 5/6 = 1/6 1/6 = $12 Total = $12 × 6 = $72”

This answer loses marks because it incorrectly adds 1/3 and 1/2 without considering 'of the remainder'. This shows a misunderstanding of the problem's structure. The working is also not clearly explained, and the final answer lacks a proper statement.

🗝️ Words that earn marks

Fraction of remainder The problem states a fraction of 'what was left' or 'the remaining amount'.
1 unit represents You are using a bar model or working backwards to find the value of one part of a fraction.
Total quantity The question asks for the original or overall amount before any parts were taken away.
Convert to improper fraction You need to multiply or divide a mixed number.
🎯

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Glossary

Key terms across this topic, in alphabetical order.

Dividend — The number or quantity that is being divided.
Divisor — The number or quantity by which another number or quantity is divided.
Improper fraction — A fraction where the numerator (top number) is greater than or equal to the denominator (bottom number), for example, 7/4 or 5/5.
Mixed number — A number that combines a whole number and a proper fraction, for example, 3 1/2.
Of (as multiplication) — In fraction problems, 'of' indicates multiplication. For example, '1/2 of 10' means 1/2 × 10.
Proper fraction — A fraction where the numerator (top number) is smaller than the denominator (bottom number).
Quotient — The result obtained when one number is divided by another.
Reciprocal — A number that, when multiplied by another number, results in 1. For a fraction, it is found by flipping the numerator and denominator.
Remainder — The amount or quantity that is left after a part has been used, spent, or taken away from an original total.
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